Wave form analyzing method and apparatus



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3,009,106 WAVE FRM ANALYZING METHOD AND APPARATUS Kurt I-I. Haase, Watertown, Mass., assigner to the United States of America as represented by the Secretary of the Air Force Filed Sept. 18, 1959, Ser. No. 840,981 12 Claims. (Cl. 324-77) (Granted under Title 35, US. Code (1952), sec. 266) 'Ihe invention described herein may be manufactured and used by or for the United States Government for governmental purposes without payment to me of any royalty thereon.

'Ihis invention relates to analysis of electrical signals in accordance with their harmonic content and more particularly to a novel method and apparatus for obtaining any desired number of Fourier coefficients of the periodic functions which constitute a complex periodic wave.

It is well known that any periodic wave which is single valued and continuous may be represented by the sum of an infinite number of sine waves of different amplitudes and frequencies. This is represented by the following equation and is known as a Fourier series:

where Apafmdt (2) am=2f01f t 00S 2mm (a) 1 bm=2 fo) sin 2mm (4) It is therefore only necessary to obtain an adequate number of values for the coeflicients Am, am and bm (known as Fourier coefficients) to provide an accurate expression for any non-sinusoidal periodic wave.

It is a common problem in the iields of engineering, applied physics and mathematics to obtain these coeiiicients and specifically the subject invention may be used in inter alia, investigations of speech sounds, X-ray examinations of the crystal structure of materials, the eld of propagation, and seismic exploration and color matching through the medium of spectro-photometric curves.

Known methods of determining the equation of a complex wave include experimental observation, the use of oscillograms, the use of a comb iilter to filter out the harmonic frequencies and the use of synthesizers which generate component waves and vary their phase, frequency and amplitude to simulate the wave to be analyzed.

These analyzing methods are limited to a finite number of coeiiicients or by the total bandwidths of filters and synthesizers.

The present invention avoids the aforementioned limitations of known Fourier analyzers by deriving, instead of the elementary natural coefficients defined by Fourier, periodic coefficients, each of which is multiplied by real number factors which are `functions of the number of harmonic orders desired and the number of segments the wave has ybeen divided into.

Accordingly, it is necessary to obtain only a certain limited number of periodic coeflicients `and to multiply these periodic coefficients by applicable real number factors to deliver any arbitrary number of natural Fourier coefficients.

It is therefore an object of this invention to teach a novel method of determining any desired number (with- 3,0%,106 Patented Nov. 14, 1961 out limit) of Fourier coefiicients as the product of a periodic coeicient and applicable real number factors.

Another object of this invention is the provision of novel automatic wave form analyzing apparatus adapted to be utilized for the derivation of Fourier coefficients by the novel method herein described.

Another object of this invention is the provision of computer means which will carry out said method without being restricted to any fixed number of coeiiicients.

Another object of this invention is to provide a Fourier analyzer that is adjustable to the curve to be analyzed and will lend itself to derivation of any desired number of coefficients.

Another object of this invention is to provide a methodY for computing any desired number of Fourier coeiiicients for the purpose of programming with a conventional desk computer.

Another object of this invention is to provide automatic means for computing any number of Fourier coeiiicients for a non-sinusoidal periodic wave.

In general the novel Fourier analyzer to be described herein comprises: a period recognizer; a sampling device and storage means where an adequate number of samples of the wave to be -analyzed are taken and stored to the end of the computation process; operational circuits to group said sample values according to their common harmonic order and multiply said groups by a number factor which is a function of the number of Fourier coefficients desired; and attenuators to derive the natural coeiiicients from the said periodic coefiicients.

The novel features which I believe to be characteristic of my invention are set forth with particularity in the appended claims. My invention itself, however, together with further objects and advantages thereof can best be understood by reference to the following description taken in connection with the accompanying drawings, in which:

FIGURE 1 illustrates a typical periodic non-sinusoidal wave and its approximate equivalent consisting of a polygonial sequence of straight lines;

FIGURE 2 illustrates a typical continuous oscilloscope trace such as is produced in the period recognition of a non-sinusoidal wave.

FIGURE 3 illustrates the period recognizing stage and sampler switch of one embodiment of my invention;

FIGURE 4 illustrates collector switches and phase converters of the first grouping stage of the invention.

FIGURE 5 illustrates adder means and sampler switches of the first grouping stage of the invention;

FIGURE 6` illustrates the multiplier stage of the invention;

FIGURES 7 7a, 7b and 7c illustrate the program switch array of the second grouping stage of the invention.

FIGURES 8 and 8a illustrate the sampler switches, phase converters and adder means of the second grouping stage of the invention.

FIGURE 9 illustrates the collector switches of the second lgrouping stage of the invention;

FIGURE l0 illustrates the phase meter and root mean square meter stage of the invention;

FIGURE ll illustrates the attenuator stage of the invention; and

FIGURE l2 illustrates a block diagram of the component parts of the invention.

Referring now to FIGURE l the period of a function TU) (dotted) which is in suiiicient correspondence with the polygonial sequence of straight lines 1"(t) (solid lines) is shown to be divided into an integral number of segments in the t (time) direction. If `a periodic function f(t) is approximated by a polygonial trace of straight lines, then the Fourier coeiicients am and bm are approximately the same for f(t) and for its approximation. I

have found that if the corners of the polygon trace are equidistant in the direction of the l-axis, so that the interval of the period is divided into n equal sections, then the coefficients can be written as products.

symbol intended to differentiate the periodic functions comprehended by my invention from the so-called natural Fourier functions. rN is an integer defined as 7L amzKnCmam* (5 5 N: bm=KnCmbm' (5) Capital letters H and F are summations of 4 distinct samwhere Kn is defined as a number factor dependent upon ple Values yb iVm-1 ynw and. iVm-r* Cap'lta H 1S the number of equal timewise segments f(t) has been di- Cofllated t @Veil fmdexed Coeclenfs 022 E pltal F (0 vided into, Cm is a number factor which is tabulated for 10 Odd indexed COHCIBDS any harmonic order m as illustrated in Table I; and am* a'n-l-l k and .bm @Present gl'OuPWlSe lltfcofflated Perlodlc C0' According to 1nd1ces 21 or 21-1 H and F makes a choice llcleqts' Thee paramets a`re further defined by the between group summations beginning with even or odd `0 Owing equamms and ta e' 15 numbered samples )121 or y2i 1. For computation of bm* sin 7|./4, N coeicients these capitals yare marked by a bar. Capital KDCTV 3N (7) letter G is used in surnmations involving even indices 2i in H `and P as Well as in the trigonometric function. Capi- Sin ,rm/4 N tal letter IU -is used where 2i1 appears instead of 2z'. m=- m.sin 7F 4 N (8) 20 The indices of G `and U may be, corresponding to m, 2,11 or 21H-l. Again barred capitals and are used for k i1=4N Tm. computation formulas for bm* coeicients. am yi cos Z-Nr 7=1 (9) F1= (y1-Hymn) (y2N1+)2N+1) (11) i=4N Wm 25 H1=(y1+y4N-i) -l- (LV2N1+12N+1) (12) b 6: sin

10 m y 2N1 F1= (Y1-MNA) +(y2N-1r-J2N-f-1) (13) Table of factors CD12 for parameter N=9 i=(y1y4N1)-(yawn-33ml) (14) m 0,32 m Cm? m C'm m (7m2 Using H1 `defined in (12) we get I have discovered, and it is fundamental to my inven- 1 tion, that periodic coeliioients am* and bm* are, under a0 WMD-bym) +9219 certain conditions, common parameters of intercorrelated 1 groups of periodic functions which constitute the Wave to 'V` be analyzed. More specifically I have found that in a +i=1 [H21`1'PH2] 1f N Oddzz'Y-l periodic Wave which has been segregated into n tmewise (14a) equidistant segments, as illustrated in FIGURE 1, the periodic coefficients for intercorrelated groups of periodic 1 functions yare identical `at points i, qr-z, 1r|-i and 21r-. az] (yNi'/SIIVZN This is true `also of periodic coefficients at points 2z', vr-2, 1r-{2i and 21r-2z' `and for other intercorrelated groups up to N i. Therefore, to solve for any number of Fourier coefficients it is only necessary to obtain n sample y values of the Wave to be analyzed, group said sample y values as indicated and multiply the groups by the number factors Kn and Cm. To facilitate grouping of values, the following equations have been formulated in which the notations involving group summations have a systematic background in the choice of capital letters and indices thereby making their distinction easy. Coefficients, and related functions thereof, associated with said intercorrelated groups of periodic functions are desig- (14b) It can be shown that if lm (N- 1) Fain-m (15) aN-m=llN+m (150.) b3= -btN-m (16) bN-m: -bN-i-m (16a) nated by an asterisk (f), said asterisk being a general Introducing the sums defined in (11) (14) into Equations 9 and 10 We deiine now two subgroups of sums, namely where only odd indexed sums F214, H214, F214 and 21 1 are included.

Furthermore we discriminate between even indices me=2p. and odd indices m0=2,rr-1.

S0, We define, if N odd: Zfy-l Substituting the subsums Gm and Um into Equations 9 and 10 we get and if N odd=2y-1 and if N even=2'y N-(zn-r) bNHmr-l) (il/N II/aN) 1) (UW-1"- Gbr-1) (21a) By Equations 19, 20 and 21 all periodic coe'icients are known.

As it has been already stated, Fourier analysis can be carried out by using any number of timewise equidistant samples within one period of the periodic function to be analyzed. To use n=4N samples where N is an integer has practical advantages. For the design outlined in the following paragraphs n is assumed to be 36 and N consequently equals 9.

Based on theory and example the Fourier analyzer has 3 parts:

Part 1 comprehends a Period Recognizer and a Storage Device where each of the 36 samples on FIG. l (y0-0, y1, y2 3135, )136:0) is stored up to the end of the cornputation process.

Part 2 comprehends all the Operational Circuits to compute the periodic coefficients. The operations involved are sampling and collecting of values derived from the sample values y0 yas, additions and polarity conversions and finally multiplications of sampled or derived values by real values l.

Part 3 comprehends attenuators to derive the natural coeicients from the periodic ones.

Naturally many variations in the performance of the computation process are possible. For instance an addition of values can be carried out simultaneously@ in time sequence. Switching devices can be combined to minimize the display of circuitry. So the description of the `apparatus has to be taken as one example to perform the general principle based on what is described in the paragraph theory.

As far as switching devices are used they are represented as mechanical rotary switches with one rotation arm resting on concentrically arranged contacts. They can always be replaced by suitable electronic switching circuits. Additions are made on the basis of simultaneity. The values in operation may be continuous, quantized or digitized. No attempt is made to achieve minimization. If the period of U) (as defined in Equation 1) is not known, it can be recognized by a Lissajous figure written on the screen of an oscilloscope. As shown in the block diagram of FIG. 3 a sine wave sweep oscillator covering a frequency range in which the yfundamental frequency 1/ T of TU) can .be expected, is linked to the y-plates of an oscilloscope. The function (t) to be investigated iS linged to the x-plates. By sweeping the oscillator frequency la steady picture will be received on the screen if one of both frequencies (oscillator frequency and frequency of (t)) is an integer multiple of the other. If the oscillator frequency is identical to 1/ T, the picture will be a closed loop without crossings as shown in FIG. 2 using TU) in FIG. l. In case of multiples, loop crossings will appear. As soon as a picture as shown in FIG. 2 Will appear, the instantaneous sweep frequency of the sweep oscillator represents the desired fundamental frequency of TU). As shown in the example in FIG. l subsequent zero crossings are not apt to derive the fundamental frequency. But since a group of zero crossings is in periodic repetition, the computation process can be started with any zero crossing of Rt). To start with a zero crossing is convenient but no necessity. The start pulse to begin the computation by sampling the values y y36 is given by the first zero crossing after the frequency recognition. This means that the Sampler Switch S1 in FIG. 3 starts and within the time T of one period it samples the values y0 y25 and stores them on its contacts. Only one revolution of the switch is necessary. This is important in so far as the duration of the existence of the periodic function T0) may be short (as in the case of vowel analysis in speech for instance).

As the next step the operations given by Equations 1l 14 have to be performed.

There is a Switch Box 1 shown in FIG. 4 containing 4 collector switches S2, S3, S1 and S6. Each collector switch has 8 contacts connected with the corresponding contacts on sampler switch S1 in FIG. 3. Each switch is connected to contacts 1 `4, switch S2, S3 and S5 in parallel to a converter each and then to contacts 2', 3 and 4.

The contacts 1 4 and 2' 4' are repeated on FIG. 5. This figure shows a Switch Box 2 containing 4 position collector switches S6, S1 and S2. The positions are marked by F, H, and Each collector switch is connected to and Adder apt to add 4 simultaneous values. Its output is connected to Sampler Switch S9 with 4X8=32 positions or contact.

Assume S6, S, and S8 in position F. During one revolution of switches S2 S5, the adder gets to its input exactly the 4 elements in Equation 11 while i runs from 1 to 8. The sums will be stored on contacts F1 F8 by switch S2 which makes in the same time a quarter revolution. Naturally S2 S5 and S6 S2 always have to be interphased and synchronous. For the next revolution of S2 S5 and the next quarter revolution of S2 switches S6 S6 go into position H and Equation 12 is performed. Similarly Equations 13 and 14 will be performed in positions Vand of S6 S2. So, in 4 revolutions of S2 S5, 4 positions of S6 S3 and one revolution of S9 all sums given by Equations 1'1 =14 will be stored on the contacts of S9.

Now we have to build the sums written down in Equations 17, 17a, 17b, 17e, 17d, 17e, 17j and 17g. T'his means that we have to multiply at first the already stored values F1, H1, F1 and E1 by the real coeicients c6=1, c1 ca enumerated in table. Since the coeiiicients c1 c6 are numbers smaller than 1, the multiplication can be performed by a potentiometer with tap ratios oorresponding to the coeflicients c1 c2 while c0=1 is the value itself. On FIG. 6 we find a collector switch S10 with contacts parallel to those of S9 connected to another collector switch S11 where the contacts are interconnected by resistors R1 R2 and the last contact is linked to ground (=0) over resistor R2. Let S16 for instance be in position F1 while S11 makes one revolution, S11 delivers in time sequence all products F1.l F1os. If so S10 keeps each position for the duration of one revolution of S11, sampler switch S12 receives 288 products. Of these products only 144 are used which is a consequence of the fact that for instance no combination of even indexed F21 with odd indexed c21-|1 is used. 144 sample values are selected by S12 and linked to the array of a program switch shown in FIGS. 7 7c. This program switch has 8 parts. Each `works so that the array of contacts is brought in 4 or 5 positions for a summation operation. PS2 and PS2 in FIG. 7 results in G0, G2, G4, G6 and G8, in G1, G2, G5, G1 and G9. In the same way PS3 and PS4 in FIG. 6c result in U0 U2 (U9=O). Also in FIGS. 7b and 7c PS5 and PS2 result in 1 9 (F-.0) and PS6 and PS7 result in G1 @a (0='(9=0). The sums G, U, and are stored and linked to the contacts of the collector switches S12 S16 in FIGS. 8 and 8a. Going on in the example we are now able to compute the periodic coefficients a* and bi according to the formulas given in Equations 19 19e and 2O and 20a. As an essential part in the summation pairs of equally indexed Gs and Us, 's and s have to be combined in addition and in subtraction. FIG. 8 shows a pair of collector-switches both in phase and synchronism. Collecting all Gs and Us in one revolution these values are linked to 2 inputs of the left Adder, while the Us in parallel are converted in their polarity and with the Gs linked to 2 inputs of the right Adder. According to Equations 19 and 20 a sum combination of the original samples yg, y12 and )127 has to be added to get the periodic coeicients. These samples are available from the switch contacts of switch'S1 on FIG. 3 and it is evident from earlier descriptions how to get the combinations i(y2iy27) +3112 and so on. These combinations are linked to contact groups of the c01- lector switches S12 and S12 in FIG. 8. These collector switches are in phase and synchronism with the upper ones and their outputs are linked to the third input of the Adders. During one revolution of the collector switches and Adders deliver the ai" coeiiicients to sampler switches S21 and S22 in FIG. 9. FIG. 8a in which the automatic computation of the b* coeicients is shown is evident from the preceding description. The results are linked to the sampler switches S22 and S24 in FIG. 8. The periodic phase angles m"=tg1 (bnf/auf) and the periodic amplitude coefcients am*=\/am*2lbm"2 can now be obtained. FIG. 10 shows 2 collector switches with the periodic coefficients a* and b* arranged on their contacts. The operation will be done by a Phase Meter and a Root Mean Square Meter respectively which also are well known operators. The resulting phases 49" and periodic root mean square amplitudes A* are distributed by sampler switches S22 and S22.

Nothing further must be done to` get the natural phases 1b since they are identical to the periodic ones. To get the natural amplitudes `from the periodic ones the periodic coeflicients A* have to be multiplied by real values resulting from the Table of Factors. But, since these values again are smaller than 1, the multiplication again can be performed by resistor potentiometer or attenuators with proper tapping. The periodic amplitudes are linked to such attenuators as shown in FIG. 11. Each attenuator has its own tapping resulting from the Table of Factors. The other end has ground potential of course. It can be tapped as high `as wanted. In FIG. 11 all natural amplitudes up to the th can be received. If the natural coefficients am and bm are wanted, the collector switches S25 and S26 in FIG. 10 `are linked directly to the potentiometers `in FIG. 11. In both cases the final result appears at the taps` of the potentiometers shown in FIG. 11 and can be visualized in any known method. 

